Let $g:\mathbb{R}^n\times \mathbb{R}^{n'}\times [0,1]\rightarrow \mathbb{R}^n$ a continuously differentiable function. Can we deduce that $g$ is locally Lipschitz ?
which is equivalent that for all nonempty compact sets $K\subset\mathbb{R}^n\times \mathbb{R}^{n'}\times [0,1] $, there exosts $M>0$ such that $$\| g(x_1,y_1,t)-g(x_2,y_2,t) \| \leq M(\|x_1-x_2 \| + \|y_1-y_2 \| ),$$ for all $(x_1,y_1,t),(x_2,y_2,t) \in K$ ?
Or should we assume some additional convexity hypothesis ?
Yes, we can already deduce local Lipschitzianity. Just note that $$ \sup_{(x, y, t) \in \overline{B} \times J} \lVert Dg(x, y, t) \rVert < \infty $$ for some ball $B \subseteq \mathbb{R}^n \times \mathbb{R}^{n'}$ and interval $J$ such that $B \times J \supseteq K$ because of domain compactness and continuity of $Dg$. We can choose such ball and interval because $K$ is bounded.
Then use the Mean Value Theorem to follow Lipschitz-continuity on $\overline{B} \times J$ which implies Lipschitz-continuity on $K$.